fdtd modeling of dispersive bianisotropic media using z-transform method

2268 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 6, JUNE 2011

FDTD Modeling of Dispersive Bianisotropic MediaUsing Z-Transform Method

Vahid Nayyeri, Student Member, IEEE, Mohammad Soleimani, Jalil Rashed-Mohassel, Senior Member, IEEE,and Mojtaba Dehmollaian

Abstract—The finite-difference time-domain (FDTD) techniquefor simulating electromagnetic wave interaction with a dispersivechiral medium is extended to include the simulation of dispersivebianisotropic media. Due to anisotropy and frequency dispersionof such media, the constitutive parameters are represented byfrequency-dependent tensors. The FDTD is formulated using theZ-transform method, a conventional approach for applying FDTDin frequency-dispersive media. Omega medium is consideredas an example of bianisotropic media, the frequency-dependenttensors of which are based on analytical models. The extendedFDTD method is used to determine the reflection and transmissioncoefficients of co- and cross-polarized electromagnetic waves fromomega slabs, illuminated by normally incident plane waves. Threecases are simulated: 1) a slab of uniaxial omega medium with itsoptical axis parallel to the propagation vector; 2) a slab of rotateduniaxial omega medium with its optical axis not parallel to thepropagation vector; and 3) a slab of biaxial omega medium. Theresults are validated by means of comparisons with analyticalsolutions.

Index Terms—Bianisotropic media, chiral medium, dispersivemedia, finite-difference time-domain (FDTD), omega medium,Z-transform method.

I. INTRODUCTION

A rapidly growing research interest in the field of electro-magnetics is the physical and electromagnetic (EM) prop-

erties of materials. Due to the need for special EM properties,artificially structured metamaterials, such as bimedia, classifiedas materials exhibiting magnetoelectric coupling, are suggested.

Bimedia are investigated for various applications such asantenna radomes [1], [2], waveguides [3]–[5] and polariza-tion transformers [6]. They are also studied for substrates ofmicrostrip antennas [7]–[10], absorbers [11], [12], backwardwave media [13], and cloaking materials [14].

In these materials, there is a coupling between electric andmagnetic fields which causes simultaneous production of elec-tric and magnetic polarizations due to the electric or magnetic

Manuscript received April 13, 2010; revised October 06, 2010; acceptedNovember 08, 2010. Date of publication April 19, 2011; date of current versionJune 02, 2011.

V. Nayyeri and M. Soleimani are with the Department of Electrical En-gineering, Iran University of Science and Technology, Tehran, Iran (e-mail:[email protected]; [email protected]).

J. Rashed-Mohassel and M. Dehmollaian are with the Center of Excellenceon Applied Electromagnetic Systems, School of ECE, University of Tehran,Tehran, Iran (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAP.2011.2143677

excitation. This phenomenon is mathematically represented bymagnetoelectric coupling tensors or coefficients in constitutiverelations which make electric and magnetic flux densities de-pendent on both electric and magnetic fields.

Bimedia themselves are divided into biisotropic and bian-isotropic subclasses. The properties of the former are not depen-dent on the direction; so its constitutive parameters are scalars.However, the latter, which exhibits different behaviors in dif-ferent directions, has tensor constitutive parameters. Chiral andpseudochiral omega materials are samples of biisotropic andbianisotropic media, respectively.

Modeling EM wave interaction with bimedia is an importantproblem, which has been investigated analytically and numeri-cally. Finite-difference time-domain (FDTD) is one of the nu-merical methods used for modeling chiral media as well as bi-isotropic media. FDTD modeling of biisotropic media is studiedin different articles [15]–[25], in some of which the effects offrequency dispersion are accounted for [20]–[25].

Many approaches are introduced for modeling dispersivemedia using FDTD method. Several summaries of variousmethods are given in [26]–[28]. Among them, the auxiliarydifferential equation (ADE), the Z-transform, and the piecewiselinear recursive convolution (PLRC) approaches are widelyused and proven to be very accurate. Comparisons of their ac-curacy and stability, given in [29] for Lorentz media, show thatthe stability limits of these approaches are approximately thesame. However, the Z-transform approach has the best accuracynear resonant frequencies and is very easy to implement.

Demir et al. [22] studied FDTD modeling of chiral mediausing the Z-transform method. They incorporated the dispersivenature of permittivity, permeability and chirality in the FDTDformulation and provided a full dispersive model in which fre-quency dependence of permittivity and permeability follows theLorentz model and that frequency dependence of chirality fol-lows the Condon model. Their work is based on convertingfrequency domain constitutive relations to Z-domain relations,proper forms for deriving FDTD updating equations. In thistransform, backward time differences are used for time deriva-tives in order to decouple updating equations for electric andmagnetic field components. Another FDTD model of dispersivechiral media using transformation to Z domain was developedby Pereda et al. [25]. In [25], Maxwell’s curl equations, firstlyexpressed in Laplace domain, are directly transformed to dis-crete-time domain. On the other hand, the constitutive relations,also expressed in Laplace domain, are first converted to Z do-main using Mobius transform [30]; they are then expressed inthe discrete-time domain. In this method, central differences are

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NAYYERI et al.: FDTD MODELING OF DISPERSIVE BIANISOTROPIC MEDIA 2269

used for time derivatives to preserve the second-order accuracyof the conventional FDTD.

Furthermore, Akyurtlu et al. introduced a FDTD methodfor modeling transverse propagation through a uniaxial bian-isotropic medium [31] called BA-FDTD. This method is basedon decomposing electric and magnetic fields into the wave-fields. They treated the anisotropic chiral medium problem asthe sum of two problems associated with non-chiral anisotropicmedia. In their simulation, an axially chiral medium as a bian-isotropic medium with no frequency dispersion is considered.

Furthermore, omega medium is a bianisotropic mediumformed by including -shaped metal elements inside anisotropic dielectric material. This medium is suggested bySaadun and called pseudochiral medium [32]. When anexternal electric (or magnetic) field is applied to an omegaparticle, it induces an electric dipole moment parallel to itsstem and a magnetic moment orthogonal to its loop. This causesmagnetoelectric coupling in the medium. As mentioned before,the constitutive parameters of this medium are frequency de-pendent tensors. Several papers have introduced a number ofreliable analytical models of particles and omega mediumwhich have been developed and validated numerically andexperimentally [33]–[35].

In the present paper, the FDTD formulation for dispersivechiral media using the Z-transform method [22], simply imple-mentable and well accurate, is extended in order to include amore general case of bianisotropic media. Omega medium is as-sumed as an example of bianisotropic media whose frequencydispersion model follows the forms given in [13], based on theanalytical models. In Section II, the theoretical development ofthe FDTD formulation in dispersive bianisotropic media is pre-sented. Finally, in Section III, the presented FDTD formulationis validated by simulating the interaction of a plane wave withan omega slab in three cases of uniaxial omega slab, rotated uni-axial omega slab, and rotated biaxial omega slab. The analyticalsolution method, given in [36], is used for validating the results.

II. THEORETICAL DEVELOPMENT

In this section, the basic theory of FDTD formulation for dis-persive bianisotropic media is presented in two parts. Since con-stitutive relations of media are used in deriving FDTD updatingequations, the constitutive properties of the media are describedin part A. These relations are considered in the forms suggestedby [37] for uniaxial and biaxial cases. Next, an omega mediumis studied as a sample of bianisotropic media. The frequency dis-persion models of this medium are taken from [13]. Finally, theeffect of axes rotations of the medium is investigated. In part B,the FDTD formulation for dispersive bianisotropic media, dis-cussed in part A, is explained using the Z-transform method.

A. Constitutive Relations and Frequency Models

Bianisotropic media are classified as media with two proper-ties, anisotropy and magnetoelectric coupling.

A bianisotropic medium can be made by including smallmetal particles with loops and handles (like ) embedded in adielectric host medium. The resonant behavior of particles leadsto the frequency dispersion of the medium which makes theconstitutive parameters frequency dependent. One type of such

Fig. 1. Geometry of the omega structure: (a) a uniaxial medium with opticalaxis along �� direction. (b) A biaxial medium.

materials is a pseudochiral medium (omega medium) thatcan be realized as a composite with -shaped metal inclusions,shown in Fig. 1. In Fig. 1(a), -shaped metal elements lie inplanes parallel to xz- and yz-planes (i.e., uniaxial medium)and, in Fig. 1(b), all of these elements lie in planes parallelto yz-plane (i.e., biaxial medium). Generally, the constitutiverelations in a reciprocal bianisotropic medium in frequencydomain can be written as [37]

(1.a)

(1.b)

where , , , and are the electric field, electric flux density,magnetic field, and magnetic flux density, respectively. In (1), ,

, and present permittivity, permeability, and magnetoelectriccoupling frequency-dependent tensors, respectively. The super-script stands for transpose. It is noted that due to anisotropy ofthe medium, the constitutive parameters are in tensor forms and,because of frequency dispersion, they are frequency dependent.In the following, constitutive relations of two cases (uniaxial andbiaxial) of bianisotropic media are reviewed.

According to Fig. 1(a), in a uniaxial bianisotropic mediumin which the optical axis is along , the constitutive tensors aregiven by

(2.a)

(2.b)

(2.c)

where and are thetwo-dimensional (2-D) transverse symmetric and antisymmetric


unit dyads defined in the xy-plane, respectively. Here, subscriptstands for transverse. Also, in (2.c), is a complex dimen-

sionless parameter that measures the magnetoelectric coupling

effect. Substituting (2) in (1) and considering , theconstitutive relations in a uniaxial bianisotropic medium withoptical axis along are determined by

(3.a)

(3.b)

Moreover, in a biaxial bianisotropic medium, as shown inFig. 1(b), the constitutive tensors are given by

(4.a)

(4.b)

(4.c)

The constitutive relations of this medium are obtained by sub-stituting (4) in (1).

(5.a)

(5.b)

Considering an omega medium as a bianisotropic mediumand assuming the quasi-static polarization of particles inthe z-direction, the dispersion of relative normal permittivity,

, and permeability, , can be neglected [33]. The relativetransversal permittivity, , relative transversal permeability,

, and magnetoelectric coupling, , assuming low-density ofparticles, are given by [13]

(6.a)

(6.b)

(6.c)

where , , and are resonant frequency, damping ratio,and magnetoelectric coupling coefficient, respectively. In (6),and are static relative permittivity and permeability, andand are relative permittivity and permeability at frequenciesmuch larger than , respectively.

If the coordinate axes rotate by Euler angles , , and , theconstitutive tensors of the medium with respect to the rotatedcoordinates, , , and can be calculatedthrough

(7.a)

(7.b)

(7.c)

where , , and are parameters, given in (2) and (4) for uni-axial and biaxial media, respectively. In (7), is the rotationmatrix defined by

(8)

where , , and are Euler angles [38].Substituting (4) and (2) in (7), the constitutive tensors of ro-

tated uniaxial and biaxial bianisotropic media are calculated by(9) and (10), respectively, as follows:

(9.a)

(9.b)

(9.c)

(10.a)

(10.b)

(10.c)

In the following subsection, the extension of FDTD methodfor simulating the wave interaction with dispersive bianisotropicmedia is presented.

B. FDTD Formulation of Dispersive Bianisotropic Media

Basically, the FDTD method is based on simultaneous solu-tions of Maxwell’s curl equations in the time domain

(11.a)

(11.b)

Two major tasks are taken in a simple FDTD formulationof a dispersive medium: first, updating flux densities andby decomposing and discretizing the curl equations in time andspace, and second, deriving equations in order to update fields

and using constitutive relations.In the first step, updating equations for and are

formed by decomposing (11) into x, y, and z componentsand discretizing them in time and space. For example, for thecomponent x, we have


(12.a)

(12.b)

Components y and z can also be obtained in a similar way.In the second step, the updating equations for field values

are derived using constitutive relations given in the frequencydomain (1). Since the updating equations are time-domainrelations, the frequency-domain constitutive relations (1) maybe converted to the time-domain ones. This transformationinvolves convolution integrals. To avoid the time consumingcomputation of convolutions, the constitutive relations areexpressed in Z domain as follows:

(13.a)

(13.b)

where is the sampling period of time discretization in Z trans-form, and , , and are the Z transforms of thetime-domain tensors whose Fourier transforms are , ,and , respectively. The main advantage of using Z trans-form is that multiplications in the frequency domain are pre-served in the Z domain. This approach is suggested by Sullivanas “Z-transform method” for FDTD modeling of wave propaga-tion in dispersive media [39]. In this method, determining con-stitutive parameters in the Z domain is required. If the Z-domainconstitutive parameters are expressed in the form of

where , , , and are constants and , thenand are simply expressed by polynomials of order .It is noticed that since in the Z domain is equivalentto in the time domain, a simple updating algorithmis obtained.

Given constitutive tensors in bianisotropic medium (9) and(10) and their frequency-dependent components (6), the task isto evaluate , , and . First, by separating the fre-

quency-dependent parts from the independent parts, the follow-ings are concluded

(14.a)

(14.b)

where , , , , and are given by

(15.a)

(15.b)

(15.c)

(15.d)

(15.e)

In (15), , , and

in the uniaxial case, shown in Fig. 1(a) and

, , ,

, and in the biaxial case, shown in Fig. 1(b).To convert frequency-domain relations to Z domain, the fol-

lowing relations are used [39]:

(16.a)

(16.b)

(16.c)

(16.d)

where and are constants.Applying these transformations to (15), the Z-domain consti-

tutive tensors , , and in (13) become (17), shownat the bottom of the next page.

The next task is to express constitutive relations in Z domain.This is obtained by substituting (17) in (13)

(18.a)

(18.b)

where

(19.a)


(19.b)

(19.c)

(19.d)

(19.e)

(19.f)

(19.g)

(19.h)

(19.i)

It is interesting to note that by separating z terms, s (18) can bewritten as

(20.a)

(20.b)

where vectors , , , and are defined as

(21.a)

(21.b)

(21.c)

(21.d)

At this point, the Z-domain equations that yield the updatingequations for field values can be simply extracted from (20) inthe following way:

(22.a)

(22.b)

Now, the property of Z transform isused in the derivation of updating equations of fields and .By setting the sampling period, , equal to one-half of the timestep in FDTD [39], (22) can be written as

(23.a)

(23.b)

in discrete-time domain. Equations (23) are decomposed into x,y, and z components and used in the FDTD formulation. Forexample, for the x components of fields we have

(24.a)

(17.a)

(17.b)

(17.c)


Fig. 2. Normal incidence of an x-polarized plane wave on a dispersive bian-isotropic slab. (a) Uniaxial omega slab. (b) Rotated uniaxial omega slab. (c)Rotated biaxial omega slab.

(24.b)

where and are elements of inverse matrix of

and , respectively, indexed by . The y and z componentscan be obtained similarly.

According to updating (23), new values of and dependon and calculated in the present time step and vectorscalculated in previous time steps. The values of flux density vec-tors and are given by (12) for the component x. However,the updating equations for vectors are required. This is ac-complished using (21). Vectors , , , and are firstwritten as

(25.a)

(25.b)

(25.c)

(25.d)

By expressing the aforementioned equations in the discrete-timedomain, updating equations for vectors are simply obtainedby

(26.a)

(26.b)

(26.c)

(26.d)


Fig. 3. 1-D FDTD cell used for modeling the bianisotropic media.

It should be noted that all of these equations must be decom-posed into components x, y, and z in order to be used in FDTDformulation. For example, for the x components of vectors,the following can be presented

(27.a)

(27.b)

(27.c)

(27.d)

According to (23) and (24), the vectors and must becalculated at points , , and

. However, the vectors and must be evaluatedat points , ,and . As a result, all components ofthe electric and magnetic fields must be known at different sixpoints while in a conventional Yee’s cell, each component ofa field vector is defined only at one point. For instance, the xcomponent of electric field is defined at . Toovercome this problem, each component of the field is defined atfive undefined points as the average of its values at neighboringdefined points. The average formulas are given in the Appendix;for example, for the component x of the electric field.

At this point, the algorithm of FDTD is completed. Theprocess of the FDTD in a dispersive bianisotropic medium iscarried out in the following order:

1. Updating all components of flux vectors [using (12)];2. Updating all components of field vectors [using (24)];3. Defining all components of electric and magnetic field

vectors at undefined points as the average of their valuesat neighboring defined points (see Appendix);

4. Updating all components of vectors [using (27)].

In the next part of the paper, this algorithm is applied to theproblem of normal incidence of electromagnetic waves, on anomega slab in several cases of arrangement of -shaped parti-cles in the slab.

III. FDTD SIMULATION OF NORMAL INCIDENCE ON A SLAB

OF BIANISOTROPIC MEDIUM

In this section, the presented formulation is used to simulateplane wave normal incidence on a dispersive bianisotropic slabin three different cases illustrated in Fig. 2. First, the slab is as-sumed to be a uniaxial medium where the optical axis is alongthe propagation vector [Fig. 2(a)]. Second, a rotated slab of uni-axial medium where the optical axis is not along propagationvector is considered [Fig. 2(b)]. Third, a slab of rotated biaxialmedium is simulated [Fig. 2(c)]. As a numerical example in oursimulations, a slab of omega medium is considered through thefollowing parameters

(28)

It is noted that with these values, constitutive tensors obey theconditions for constitutive tensors of lossy bianisotropic media,given in [40].

The thickness of the slab is assumed to be 10 cm (in thedirection). The 1-D FDTD computational space is 1 meter longwhich consists of 1000 cells where the slab is assumed to belocated between cells 450 and 550 in the middle of computa-tional space. A schematic of 1-D FDTD cell is presented inFig. 3. As noted in Section II, attendance of each componentof the electric and magnetic fields is necessary for FDTD mod-eling of Bianisotropic media. The soft lattice truncation condi-tions [41] are applied on both sides of computation space. Be-cause of dispersion and bianisotropy of the medium, the timestep should be chosen short enough to avoid the instability ofFDTD method. Hence, in this paper, the time step is set to

, where is the length of cells,is the slab thickness, and is light velocity in free space. The

choice of very small is due to the special frequency disper-sion model (6.b). In FDTD simulations, the slab is illuminatedby an x-polarized Gaussian waveform of electromagnetic planewave propagating along . It is desired to transform time-do-main results of FDTD to frequency domain and calculate thetransmission and reflection coefficients. Discrete Fourier trans-form (DFT) is used to obtain frequency-domain data. The re-sults of FDTD method are compared with the exact results, cal-culated based on the analytical method presented in [36]. Thecomparison of results illustrates the high accuracy of the pre-sented FDTD method.

A. Normal Incidence on a Uniaxial Omega Slab, PropagationAlong the Optical Axis

In the first example, a slab of uniaxial omega medium with anoptical axis along is assumed which is illuminated by an x-po-larized plane wave propagating in the direction [Fig. 2(a)]. Theconstitutive relations of medium are given in (5.a) and (5.b) and


Fig. 4. Simulation results of normal incidence of a Gaussian plane wave onuniaxial omega slab along its optical axis. (a) Co-polarized reflected field in timedomain. (b) Co-polarized transmitted field in time domain. (c) Co-polarizationof reflection and transmission coefficients in frequency domain.

constitutive tensors and their components are given in (6) and(28).

From (3), the bianisotropy of the medium couples the x andy components of electric field to the y and x components of themagnetic field, respectively. As the incident wave consists of xand y components of electric and magnetic fields, respectively,other components of fields, component y of the electric fieldand component x of the magnetic field, do not appear. Hence,the cross-polarizations of reflected and transmitted waves donot exist. The FDTD simulation agrees with this statement andcross-polarized reflection and transmission coefficients are zero.In Fig. 4, the co-polarizations of FDTD simulation results areshown and compared with analytical results. The comparisonshows a good agreement between the developed FDTD resultsand analytic solutions. Time-domain reflected and transmittedwaves are presented in Fig. 4(a) and Fig. 4(b), respectively, andfrequency-domain reflection and transmission coefficients aredemonstrated in Fig. 4(c).

B. Normal Incidence on Rotated Uniaxial Omega Slab

In the second example, a normally incident x-polarized planewave propagating along the direction illuminates a slab ofuniaxial omega medium with its optical axis not parallel to thepropagation vector [Fig. 2(b)]. For this case, rotated particlesare assumed in the slab. For instance, the Euler angles are setto , , and . The constitutive tensorsof the rotated medium are given by (9) and the parameters aredetermined by (6) and (28).

Due to the effect of rotation matrix in (9.c), the magneto-electric coupling tensor becomes a dense matrix with manynon-zero elements, resulting in a relatively strong magnetoelec-tric coupling between all the components of electric and mag-netic fields. Here, all components of electric and magnetic fieldsappear in the slab. Due to the anisotropy of the medium,

and . Therefore, the z components of the fields mayexist in the slab and propagate along the direction. However,these components vanish at the slab boundaries and do not prop-agate outside the slab. In this example, the reflected and trans-mitted waves have both co- and cross-polarizations illustratedin Fig. 5 and Fig. 6, respectively. The FDTD results and relatedanalytical solutions are compared. They reveal a good accuracyof the developed FDTD method for the dispersive bianisotropicmedia.

C. Incidence on Rotated Biaxial Omega Slab

In the final example, a normal incidence of plane wave on aslab of rotated biaxial omega medium is considered [Fig. 2(c)].In this case, constitutive relations are given in (10) and param-eters are defined as the previous example. Because of rotation,as in the last example, is a dense matrix. Therefore, co- andcross-polarizations of transmitted and reflected waves bothexist. The results of FDTD simulation and comparison withthe exact solution are provided in Fig. 7 and Fig. 8 for co- andcross-polarizations, respectively.

IV. CONCLUSION

In this paper, the FDTD method is developed for simulatingthe electromagnetic wave interaction with dispersive bian-isotropic media. The Z-transform method is used in developing


Fig. 5. Simulation results of co-polarized (a) reflected field. (b) Transmittedfield in time domain. (c) Reflection and transmission coefficients in frequencydomain of a rotated uniaxial omega slab illuminated by normal incidence of aGaussian plane wave.

the technique. Due to anisotropy of media, tensor notation isused in the formulation of FDTD. The developed FDTD methodis applied to simulate the interaction of electromagnetic plane

Fig. 6. Simulation results of cross-polarized (a) reflected field. (b) Transmittedfield in time domain. (c) Reflection and transmission coefficients in frequencydomain of a rotated uniaxial omega slab illuminated by normal incidence of aGaussian plane wave.

waves with both uniaxial and biaxial omega media in order toextract co- and cross-polarized, transmitted and reflected waves


Fig. 7. Simulation results of co-polarized (a) reflected field. (b) Transmittedfield in time domain. (c) Reflection and transmission coefficients in frequencydomain of a rotated biaxial omega slab illuminated by normal incidence of aGaussian plane wave.

in time domain and reflection and transmission coefficients infrequency domain. The numerical results of FDTD are com-pared with analytical results and a good agreement is observed.

Fig. 8. Simulation results of cross-polarized (a) reflected field. (b) Transmittedfield in time domain. (c) Reflection and transmission coefficients in frequencydomain of a rotated biaxial omega slab illuminated by normal incidence of aGaussian plane wave.

APPENDIX

The component x of electric field at points ,, ,


, and is not defined in a Yee cell.To define this component of the field, the same component ofthe field at the neighbors is averaged as follows:

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

Other components of and fields are averaged in a similarmanner.

ACKNOWLEDGMENT

The authors would like to gratefully acknowledge the con-structive comments of reviewers.

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Vahid Nayyeri (S’08) was born in Tehran, Iran, in1983. He received the B.S. degree from the Iran Uni-versity of Science and Technology (IUST), Tehran,in 2006 and the M.S. degree from the University ofTeheran, in 2008, both in electrical engineering.

Since 2008, he has been working toward thePh.D. degree in electrical engineering at the IUST.His research interests include finite-differencetime-domain method, electromagnetics of complexmedia, numerical analysis of electromagnetic waveinteraction with them, and optimization methods in

electromagnetic problems.

Mohammad Soleimani received the B.S. degreein electrical engineering from the University ofShiraz, Shiraz, Iran, in 1978 and the M.S. and Ph.D.degrees from Pierre and Marie Curio University,Paris, France, in 1981 and 1983, respectively.

He is working as a Professor with the Iran Univer-sity of Sciences and Technology, Tehran, Iran. Hisresearch interests are in antennas, small satellites,electromagnetics, and radars. He has served in manyexecutive and research positions including: Ministerof ICT, Student Deputy of Ministry of Science,

Research and Technology, Head of Iran Research Organization for Science andTechnology, Head of Center for Advanced Electronics Research Center; andTechnology Director for Space Systems in Iran Telecommunication Industries.

Jalil Rashed-Mohassel (SM’07) received the M.Sc.degree in electronics engineering from the Universityof Tehran, Tehran, Iran, in 1976 and the Ph.D. de-gree in electrical engineering from the University ofMichigan, Ann Arbor, in 1982.

Formerly, he was with the University of Sistan andBaluchestan, Zahedan, Iran, where he held severalacademic and administrative positions. In 1994, hejoined the University of Tehran where he is doingteaching and research as a Professor in antennas, EMtheory, and applied mathematics.

Dr. Rashed-Mohassel served as the academic Vice-Dean, College of Engi-neering, General Director of Academic Affairs, University of Tehran, and iscurrently Chairman of the ECE Department, Principal Member of Center ofExcellence on Applied Electromagnetic Systems, and the Director of the Mi-crowave Laboratory. He was selected as the Brilliant National Researcher by theIranian Association of Electrical and Electronics Engineers in 2007, and was theDistinguished Professor (2008–2009) in the 1st Education Festival, Universityof Tehran.

Mojtaba Dehmollaian was born in Iran in 1978.He received the B.S. and M.S. degrees in electricalengineering from the University of Tehran, Tehran,Iran, in 2000 and 2002, respectively. He received theM.S. degree in applied mathematics and the Ph.D.degree in electrical engineering from the Universityof Michigan, Ann Arbor, in 2007.

Currently, he is an Assistant Professor with theDepartment of Electrical and Computer Engineering,University of Tehran. His research interests are ap-plied electromagnetics, radar remote sensing, and

electromagnetic wave propagation, and scattering.Dr. Dehmollaian was the recipient of the IEEE Geoscience and Remote

Sensing Symposium (IGARSS) 2007 Prize Paper award.

fdtd modeling of dispersive bianisotropic media using z-transform method

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